The Method Of Characteristics (MOC) has been widely used for two dimensional lattice calculations. One of the main drawbacks of the MOC is its poor spatial representation of the within-group energy source, which requires the use of a very fine mesh to adequately resolve the neutron flux solution. The most straightforward way to improve the spatial representation of this source would be to project the scalar flux to a set of higher-order trial functions within each mesh element, but this is expensive. An alternative to this has already been proposed that exploits the angular flux moments on the surfaces of each mesh element. In this paper, we present a new alternative in which we define a higher order representation of the isotropic source. The key feature of our method is the introduction of a new, linear discontinuous source representation that does not resort to an expensive spatial projection. This is accomplished by using a P1 approximation to construct the average derivative of the spatial source. So far, the derivation has been done only for a linear representation of the isotropic sources in two and three-dimensional geometries. The paper presents an analysis of the performance of the new method in simple cases and the Takeda 1 benchmarks.